Where were you when you realised there are no algebraic ways to find roots of polynomials greater than order 4? What's your favorite (set) of polynomials?
Polynomials General
>>7926746
Sure there are, you just have to go beyond radicals. This is still algebraic.
See:
https://en.wikipedia.org/wiki/Quintic_function#Beyond_radicals
*Implying that radicals don't exist
So there is no way to find the root of [math]x^7-1=0[/math] ?
i was dueling with evariste when abel call
"no radical formulas"
"yes"
"no"
>>7926746
>there are no algebraic ways to find roots of polynomials greater than order 4
You mean using radicals?
>order 4
You mean degree 4?
Cmon anon
Who cares ? What does "finding roots" mean anyway ? Why is a solution by radicals be satisfying ? After all, by expressing a solution with radicals you are basically saying "I cannot solve this equation, but if I could solve [math]x^n = \delta[/math], then I could solve this equation". But what makes the polynomials [math]X^n - \delta[/math] so special ?
>>7927580
>After all, by expressing a solution with radicals you are basically saying "I cannot solve this equation, but if I could solve [math]x^n=\delta[/math], then I could solve this equation"
We're perfectly capable of finding nth roots. You seem to have completely misunderstood something.
Can you please post some fancy graphics related to polynomials?
>>7927590
How ?
Prove it
>>7926746
The Jones polynomial
>>7926746
question sci, what is the best polynomial basis to interpolate noisy data. I can do a least squares fit on 100 data points to find the best 4th order polynomial using monomial basis. Is there a better basis to use? for example the lagrange polynomial will overfit the first few datapoints, laguerre basis seems a little better...
disclaimer am mechanical engineer major
>>7926746
We didn't even exist at all when your polynomials of degree 5 or more were were discovered to have no general method for solving for their roots.
>>7926746
I haven't gone into Galois theory yet, but I did derive the cubic and quartic in threads on here some months ago. I have the paperworks tucked away in my drawers.
In the course of doing so, I remember reading that /certain/ quintics could be solved once you tack on xyz, or allow pqr. But according to what I read at the time (wiki), this doesn't fit the identical notion of an /algebraic solution/ which is common to the constant, linear, quadratic, cubic and quartic cases. In these cases IIRC you may avail yourself of six operations: the big four arithmetic operations, integer exponentiation, and integer (square root, cube root etc) root extraction, but those six are IT. They define 'algebraic solutions' themselves. To add-on to these is all well and good, but it ceases to be an algebraic solution.
>>7926777
I appreciate that one "uses algebra", "uses algebraic techniques", etc, to solve quintics. However, according to what I've read, although /certain quintics/ have algebraic solutions, there does not exist a /general algebraic solution/ for the /general quintic/, which is the point of Abel-Ruffini-Galois. And, this particular, defined phrase /algebraic solution/ is what we are concerned with. As I've said, it has a rather limited and prosaic context of the six above operations, plus the boilerplate about variables etc.
I don't know about Bring radicals so I will not comment much further, except to say that the language of your link (and the ultraradicals link) is fairly clear that these pertain to /certain/ quintics, and not to a /general quintic solution/, though those certain solutions may perhaps be algebraic solutions in the sense that I mean. As for the rest, that appears to veer further and further away from our "high-school-tier" allowed algebra, above.
https://en.wikipedia.org/wiki/Algebraic_solution
>>7932404
>>Complaining about anime on a site whose culture revolves around Japanese animation
>>7927476
>poly
tell me about
[math] \displaystyle f(x) = \sum_{n=0}^{\infty} \frac{x^n}{2^n} [/math]
>>7928570
>Can you please post some fancy graphics related to polynomials?
Yes Anon.
>>7933087
[math] f(x) = \frac{1}{1-\frac{x}{2}}[/math]
>>7932479
that is a polynomial, do you even math?
>>7926746
>realised
You mean when I read it? Most people suss out very few of the laws of mathematics on their own, especially when you get past basic algebra. To answer the question I read it when I was in junior high in a book a friend of my family gave me, and I felt kind of sad about it.
>>7933114
>
>>7933575
>x^7
>-1
Two terms
>>7933583
>a number is an algebraic term
Just stop posting
>>7932430
Not him but Bring radicals can indeed find general solutions to quintic and even higher order polynomials, it's just that the solutions aren't algebraic.
>>7926746
>Take n>4 degree polynomial
>Take a linear approximate to turn it into a 4 degree polynomial
>Find roots for the 4 degree polynomial
EZ PZ
>>7927476
[eqn]
x^7 - 1 = 0\\
(x - 1)(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) = 0
[/eqn]
And now I'm stuck.
>>7933687
>it's just that the solutions aren't algebraic
Disregard this part, I'm drunk.
>>7933597
It is
>>7933729
I don't give a shit, it's a formal power series.
>>7933729
it would be extremely undefined
>>7933741
Try x^2-1, x^3-1 and x^4-1 first. See if you can generalize
>>7933597
it's -1 with an implied x^0
just a number by itself is a monomial
there are two terms in x^7 - 1
I really can't believe you're still defending yourself
Shit I learned about in my DE class.
https://en.wikipedia.org/wiki/Legendre_polynomials
>>7928570
>>7933107
I've been searching for ways of reproducing art like this. Does anyone have experience? I have some low resolution illustrations that a prof at my uni has generated.
>>7934089
find a cool family of polynomials
pick one
plot its complex roots in python or whatever
fuck with the color till is looks sexy
that's what the prof who made that blue one did anyways, afaik
>>7934468
slight correction, but still you fuck with shit until it looks cool preddy much.
> The Kazhdan-Lusztig polynomials are a particularly rich family of polynomials arising from representation theory and the geometry of the flag manifold that I am attempting to understand combinatorially. This image depicts the reciprocals of the roots of the 726,636 distinct Kazhdan-Lusztig polynomials associated to the exceptional Coxeter group H_4. The positive x-axis is pointing directly up. Shading is derived from the density of roots.
>>7933741
x^7-1=0 has 1 real and 6 imaginary solutions
>>7933729
no
[math] \displaystyle f(2) = \sum_{n=0}^{\infty} \frac{2^n}{2^n} = \sum_{n=0}^{\infty}1[/math]
[math] \displaystyle \sum_{n=0}^{\infty} 1 = 1 + \sum_{n=1}^{\infty}1 [/math]
[math] \displaystyle 1 + \sum_{n=1}^{\infty} 1 = 1+ \sum_{n=1}^{\infty} \frac{1}{n^0} = 1 + \zeta(0) = 1 + (-\frac{1}{2}) = \frac{1}{2}[/math]
>>7934619
dank.
art people please post what you know about how the images were generated
>>7934635
9/10
>>7934635
what the fuck
>>7934635
is this where the -1/12 thing comes from
>>7934635
i hate this meme
>>7934619
>>7934636
That picture was taken from this site:
>http://math.ucr.edu/home/baez/roots/
It's a "picture of all the roots of all polynomials of degree ≤ 5 with integer coefficients ranging from -4 to 4."
This one is of all the polynomials of degree ≤ 24 from -1 to 1. There's a 90mb extremely high resolution picture on that website.
